Solvability of Linear Differential Systems in the Liouvillian Sense

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n dy Bi = y, Bi ∈ Mat(p, C), (2) dz z − ai Xi=1 whose coefficients Bi are sufficiently small, Yu. S. Ilyashenko and A. G. Khovansky [9] have obtained an explicit criterion of solvability. Namely, the following statement holds: There exists ε = ε(n,p) > 0 such that a condition of solvability by quadratures for the Fuchsian system (2) with kBik < ε takes an explicit form: the system is solvable by quadratures, if and only if all the matrices Bi are triangular (in some basis). In this article we propose a refinement of the above assertion in which it is sufficient that the eigenvalues of the residue matrices Bi be small (the estimate is given), and we also propose a generalization to the case of a system with irregular singular points.

2 A local form of solutions of a system near its singular points

A singular point ai of the system (1) is said to be Fuchsian, if the coefficient matrix B(z) has a simple pole at this point. Due to Sauvage’s theorem, a Fuchsian singular point of a linear differential system is regular (see [8, Th. 11.1]). However, the coefficient matrix of a system at a regular singular point may

2 in general have a pole of order greater than one. Let us write the Laurent expansion of the coeﬃcient matrix B(z) of the system (1) near its singular point z = a in the form

B−r−1 B−1 B(z)= + . . . + + B + ..., B− − 6= 0. (3) (z − a)r+1 z − a 0 r 1

The number r is called the Poincarérank of the system (1) at this point (or the Poincarérank of the singular point z = a). For example, the Poincarérank of a Fuchsian singularity is equal to zero. The system (1) is said to be Fuchsian, if all its singular points are Fuchsian (then it can be written in the form (2)). The system whose singular points are all regular will be called regular singular. According to Levelt’s theorem [15], in a neighbourhood of each regular singular point ai of the system (1) there exists a fundamental matrix of the form

e Ai Ei Yi(z)= Ui(z)(z − ai) (z − ai) , (4)

1 p where Ui(z) is a holomorphic matrix at the point ai, Ai = diag(ϕi ,...,ϕi ) is a diagonal integer j matrix whose entries ϕi organize in a non-increasing sequence, Ei = (1/2πi) ln Mi is an upper- triangular matrix (the normalized logarithm of the corresponding monodromy matrix) whose j e f eigenvalues ρi satisfy the condition j 0 6 Re ρi < 1. Such a fundamental matrix is called a Levelt matrix, and one also says that its columns form a Levelt basis in the solution space of the system (in a neighborhood of the regular singular point j j j ai). The complex numbers βi = ϕi + ρi are called the (Levelt) exponents of the system at the regular singular point ai. If the singular point ai is Fuchsian, then the corresponding matrix Ui(z) in the decomposition (4) is holomorphically invertible at this point, that is, det Ui(ai) 6= 0. It is not diﬃcult to check that in this case the exponents of the system at the point ai coincide with the eigenvalues of the residue matrix Bi. And in the general case of a regular singularity ai there are estimates for the order of the function det Ui(z) at this point obtained by E. Corel [4] (see also [6]): p(p − 1) r 6 ord det U (z) 6 r , i ai i 2 i where ri is the Poincar´erank of the regular singular point ai. These estimates imply the inequalities for the sum of exponents of the regular system over all its singular points, which are called the Fuchs inequalities:

n n p n p(p − 1) − r 6 βj 6 − r (5) 2 i i i Xi=1 Xi=1 Xj=1 Xi=1

(the sum of exponents is an integer). Let us now describe the structure of solutions of the system (1) near one of its irregular singular points. We suppose that the irregular singularity ai of Poincar´erank ri is non-resonant, 1 p that is, the eigenvalues bi , . . . , bi of the leading term B−ri−1 of the matrix B(z) in the expansion

(3) at this point are pairwise distinct. Let us ﬁx a matrix Ti reducing the leading term B−ri−1 to the diagonal form −1 1 p Ti B−ri−1Ti = diag(bi , . . . , bi ).

3 Then the system possesses a uniquely determined formal fundamental matrix Yi(z) of the form

Λi Qi(z) b Yi(z)= Fi(z)(z − ai) e , where b b

a) Fi(z) is a matrix formal Taylor series in z − ai and Fi(ai)= Ti;

b) Λbi is a constant diagonal matrix whose diagonal entriesb are called the formal exponents of the system (1) at the irregular singular point ai;

1 p j c) Qi(z) = diag(qi (z),...,qi (z)) is a diagonal matrix whose diagonal entries qi (z) are poly- −1 nomials in (z − ai) of degree ri without a constant term,

j j bi −ri −ri qi (z)= − (z − ai) + o((z − ai) ). ri

1 Ni Furthermore a punctured neighbourhood of the point ai can be covered by a set {Si ,...,Si } of ”good” open sectors with vertex at this point (which we take to be arranged in counterclock- 1 j wise order starting with Si ) such that in each Si there exists a unique genuine fundamental matrix

j j Λi Qi(z) Yi (z)= Fi (z)(z − ai) e (6)

j j of the system (1) whose factor Fi (z) has the asymptotic expansion Fi(z) in Si (see [11, Th. 21.13, Prop. 21.17]). In every intersection Sj ∩ Sj+1 the fundamental matrices Y j and Y j+1 i i b i i necessarily diﬀer by a constant invertible matrix:

j+1 j j j Yi (z)= Yi (z)Ci , Ci ∈ GL(p, C),

Λi 1 and it is understood that the logarithmic term (z − ai) is analytically continued from Si to 2 2 3 Ni 1 Si , from Si to Si , ..., from Si to Si , so that

1 2πiΛi Ni Ni Ni 1 Yi (z)e = Yi (z)Ci in Si ∩ Si . (7)

1 Ni The matrices Ci ,...,Ci are called Stokes matrices of the system (1) at the non-resonant irregular singular point ai. They satisfy the relation

2πiΛi 1 1 Ni e = Mi Ci ...Ci , (8)

1 1 where Mi is the monodromy matrix of Yi at the point ai. Indeed, the fundamental matrix 1 1 2 2 1 −1 1 2 1 −1 1 2 Yi can be continued from Si into Si as Yi (Ci ) , since Yi = Yi (Ci ) in Si ∩ Si . Fur- 2 3 3 1 2 −1 Ni ther it is continued from Si into Si as Yi (Ci Ci ) , etc. Finally, in Si it becomes equal to − Ni 1 Ni 1 −1 1 1 2πiΛi 1 Ni −1 Yi (Ci ...Ci ) . Then it comes back into Si as Yi e (Ci ...Ci ) according to (7), whence the relation (8) follows. It is also known that all the eigenvalues of any Stokes matrix are equal to 1, that is, the Stokes matrices are unipotent (see [11, Prop. 21.19] or [19, Th. 15.2]).

4 3 Linear diﬀerential systems and meromorphic connections on holomorphic vector bundles

Let us recall some notions concerning holomorphic vector bundles and meromorphic connections in a context of linear diﬀerential equations. Here we mainly follow [11, Ch. 3] or [7] (see also [3]). In an analytic interpretation, a holomorphic bundle E of rank p over the Riemann sphere is deﬁned by a cocycle {gαβ(z)}, that is, a collection of holomorphic matrix functions corresponding to a covering {Uα} of the Riemann sphere:

gαβ : Uα ∩ Uβ −→ GL(p, C),Uα ∩ Uβ 6= ∅.

These functions satisfy the conditions

−1 gαβ = gβα , gαβgβγ gγα = I (for Uα ∩ Uβ ∩ Uγ 6= ∅). ′ Two holomorphically equivalent cocycles {gαβ(z)}, {gαβ(z)} deﬁne the same bundle. Equiva- lence of cocycles means that there exists a set {hα(z)} of holomorphic matrix functions hα : Uα −→ GL(p, C) such that ′ hα(z)gαβ (z)= gαβ(z)hβ(z). (9)

p A section s of the bundle E is determined by a set {sα(z)} of vector functions sα : Uα −→ C that satisfy the conditions sα(z)= gαβ(z)sβ(z) in intersections Uα ∩ Uβ 6= ∅ . A meromorphic connection ∇ on the holomorphic vector bundle E is determined by a set {ωα} of matrix meromorphic diﬀerential 1-forms that are deﬁned in the corresponding neighbourhoods Uα and satisfy gluing conditions

−1 −1 ωα = (dgαβ)gαβ + gαβωβgαβ (for Uα ∩ Uβ 6= ∅). (10) ′ Under a transition to an equivalent cocycle {gαβ} connected with the initial one by the relations (9), the 1-forms ωα of the connection ∇ are transformed into the corresponding 1-forms

′ −1 −1 ωα = (dhα)hα + hαωαhα . (11)

Conversely, the existence of holomorphic matrix functions hα : Uα −→ GL(p, C) such that ′ ′ the matrix 1-forms ωα and ωα (satisfying the conditions (10) for gαβ and gαβ respectively) are ′ connected by the relation (11) in Uα, indicates the equivalence of the cocycles {gαβ } and {gαβ} (one may assume that the intersections Uα ∩Uβ do not contain singular points of the connection). Vector functions sα(z) satisfying linear differential equations dsα = ωαsα in the corresponding Uα, by virtue of the conditions(10) can be chosen so that a set {sα(z)} determines a section of the bundle E, which is called horizontal with respect to the connection ∇. Thus horizontal sections of a holomorphic vector bundle with a meromorphic connection are determined by solutions of local linear differential systems. The monodromy of a connection (the monodromy group) characterizes ramification of horizontal sections under their analytic continuation along loops in C not containing singular points of the connection 1-forms and is defined similarly to the monodromy group of the system (1). A connection may be called Fuchsian (logarithmic), regular or irregular depending on the type of the singular points of its 1-forms (as singular points of linear differential systems). If a bundle is holomorphically trivial (all matrices of the cocycle can be taken as the identity matrices), then by virtue of the conditions (10) the matrix 1-forms of a connection coincide

5 on non-empty intersections Uα ∩ Uβ. Hence horizontal sections of such a bundle are solutions of a global linear differential system defined on the whole Riemann sphere. Conversely, the linear system (1) determines a meromorphic connection on the holomorphically trivial vector bundle of rank p over C. It is understood that such a bundle has the standard definition by the cocycle that consists of the identity matrices while the connection is defined by the matrix 1-form B(z)dz of coefficients of the system. But for us it will be more convenient to use the following equivalent coordinate description (a construction already appearing in [3]). At first we consider a covering {Uα} of the punctured Riemann sphere C \ {a1, . . . , an} by simply connected neighbourhoods. Then on the corresponding non-empty intersections Uα ∩ Uβ ′ one defines the matrix functions of a cocycle, gαβ(z) ≡ const, which are expressed in terms of the monodromy matrices M1,...,Mn of the system (1) via the operations of multiplication and ′ taking the inverse (see [7]). In this case the matrix differential 1-forms ωα defining a connection are equal to zero. Further the covering {Uα} is complemented by small neighbourhoods Oi of the singular points ai of the system, thus we obtain the covering of the Riemann sphere C. ′ To non-empty intersections Oi ∩ Uα there correspond matrix functions giα(z) = Yi(z) of the cocycle, where Yi(z) is a germ of a fundamental matrix of the system whose monodromy matrix at the point ai is equal to Mi (so, for analytic continuations of the chosen germ to non-empty intersections Oi ∩ Uα ∩ Uβ the cocycle relations giαgαβ = giβ hold). The matrix differential 1- ′ forms ωi determining the connection in the neighbourhoods Oi coincide with the 1-form B(z)dz ′ ′ of coefficients of the system. To prove holomorphic equivalence of the cocycle {gαβ, giα} to the identity cocycle it is sufficient to check existence of holomorphic matrix functions

hα : Uα −→ GL(p, C), hi : Oi −→ GL(p, C), such that

′ −1 −1 ′ −1 −1 ωα = (dhα)hα + hαωαhα , ωi = (dhi)hi + hiωihi . (12) ′ Since we have ωα = B(z)dz and ωα = 0 for all α, the ﬁrst equation in (12) is rewritten as a linear system −1 −1 d(hα ) = (B(z)dz)hα , −1 which has a holomorphic solution hα : Uα −→ GL(p, C) since the 1-form B(z)dz is holomorphic in a simply connected neighbourhood Uα. The second equation in (12) has a holomorphic ′ solution hi(z) ≡ I, as ωi = ωi = B(z)dz. One says that a bundle E has a subbundle E′ ⊂ E of rank k

1 1 gαβ ∗ ωα ∗ gαβ = 2 , ωα = 2 , 0 gαβ 0 ωα

1 1 1 ′ where gαβ and ωα are blocks of size k × k (then the cocycle {gαβ } deﬁnes the subbundle E and 1 ′ ′ the 1-forms ωα deﬁne the restriction ∇ of the connection ∇ to the subbundle E ).

Example 1. Consider a system (1) whose monodromy matrices M1,...,Mn are of the same blocked upper-triangular form, and the corresponding holomorphically trivial vector bundle E with the meromorphic connection ∇. Suppose that in a neighbourhood of each singular point ai of the system there exist a fundamental matrix Yi(z) whose monodromy matrix is Mi, and a holomorphically invertible matrix Γi(z) such that Γi Yi is a blocked upper-triangular matrix

6 (with respect to the blocked upper-triangular form of the matrix Mi). Let us show that to the common invariant subspace of the monodromy matrices there corresponds a vector subbundle E′ ⊂ E that is stabilized by the connection ∇. We use the above coordinate description of the bundle and connection with the cocycle ′ ′ ′ ′ ′ {gαβ, giα} and set {ωα,ωi} of matrix 1-forms. The matrices gαβ are already blocked upper- ′ triangular since the monodromy matrices M1,...,Mn are (and ωα = 0), while the matrices ′ ′ ′ giα = Yi can be transformed to such a form, Γi giα =Γi Yi. Thus changing the matrices giα onto ′ ′ Γi giα and matrix 1-forms ωi onto

′ −1 −1 Γi ωi Γi + (dΓi)Γi , we pass to the holomorphically equivalent coordinate description with the cocycle matrices and connection matrix 1-forms having the same blocked upper-triangular form. The following auxiliary lemma points to a certain block structure of a linear diﬀerential system in the case when the corresponding holomorphically trivial vector bundle with the meromorphic connection has a holomorphically trivial subbundle that is stabilized by the connection.

Lemma 1. If a holomorphically trivial vector bundle E of rank p over C endowed with a meromorphic connection ∇ has a holomorphically trivial subbundle E′ ⊂ E of rank k that is stabilized by the connection, then the corresponding linear system (1) is reduced to a blocked upper-triangular form via a constant gauge transformation y˜(z) = Cy(z), C ∈ GL(p, C). That is, ′ − B (z) ∗ CB(z)C 1 = , 0 ∗ where B′(z) is a block of size k × k.

Proof. Let {s1,...,sp} be a basis of global holomorphic sections of the bundle E (which are linear independent at each point z ∈ C) such that the 1-form of the connection ∇ in this basis ′ ′ is the 1-form B(z)dz of coeﬃcients of the linear system. Consider also a basis {s1,...,sp} of ′ ′ global holomorphic sections of the bundle E such that s1,...,sk are sections of the subbundle ′ ′ ′ −1 E , (s1,...,sp) = (s1,...,sp)C , C ∈ GL(p, C). Now choose a basis {h1,...,hp} of sections of the bundle E such that these are horizontal ′ with respect to the connection ∇ and h1,...,hk are sections of the subbundle E (this is possible since E′ is stabilized by the connection). Let Y (z) be a fundamental matrix of the system whose columns are the coordinates of the sections h1,...,hp in the basis {s1,...,sp}. Then

k × k ∗ Y (z)= CY (z)= 0 ∗ e is a blocked upper-triangular matrix, since its columns are the coordinates of the sections ′ ′ h1,...,hp in the basis {s1,...,sp}. Consequently, the transformationy ˜(z) = Cy(z) reduces the initial system to a blocked upper-triangular form. The degree deg E (which is an integer) of a holomorphic vector bundle E endowed with a meromorphic connection ∇ may be deﬁned as the sum

deg E = resai tr ωi Xi=1

7 of the residues of local differential 1-forms tr ωi over all singular points of the connection (the notation ”tr” is for the trace), where ωi is the local matrix differential 1-form of the connection ∇ in a neighbourhood of its singular point ai. Later when calculating the degree of a bundle we will apply the Liouville formula tr ωi = d ln det Yi, where Yi is a fundamental matrix of the local linear differential system dy = ωi y.

4 Solvability of regular singular systems with small exponents

Consider a system (1) with regular singular points a1, . . . , an of Poincar´erank r1, . . . , rn respectively. If the real part of the exponents of this system is suﬃciently small, then the following necessary condition, of solvability by quadratures, holds. j Theorem 1. Let for some k ∈ {1,...,p − 1} the exponents βi of the regular singular system (1) satisfy the condition

j Re βi > −1/nk, i = 1,...,n, j = 1,...,p, (13)

j l and, for each pair βi 6≡ βi (mod Z), i = 1,...,n, one of the conditions

j l j l Re βi − Re βi 6∈ Q or Im βi 6= Im βi. (14)

Then the solvability of the system (1) by quadratures implies the existence of a constant matrix C ∈ GL(p, C) such that the matrix CB(z)C−1 is of the following blocked form:

′ − B (z) ∗ CB(z)C 1 = , 0 ∗ where B′(z) is an upper-triangular matrix of size k × k. Remark 1. Though the inequalities (13) restrict the real parts of the exponents from below, together with the estimates (5) they provide boundedness from above. Remark 2. The sum of the Poincar´eranks of a regular singular system whose exponents satisfy the condition (13) is indeed restricted because of the Fuchs inequalities (5), namely, n i=1 ri < p/k. P The proof of the theorem is based on two auxiliary lemmas. j Lemma 2. Let the exponents βi of the regular singular system (1) satisfy the condition (13). If the monodromy matrices of this system are upper-triangular, then there is a constant matrix C ∈ GL(p, C) such that the matrix CB(z)C−1 has the form as in Theorem 1. Proof. We use a geometric interpretation (exposed in the previous section) according to which to the regular singular system (1) there corresponds the holomorphically trivial vector bundle E of rank p over the Riemann sphere endowed with the meromorphic connection ∇ with the regular singular points a1, . . . , an. Since the monodromy matrices M1,...,Mn of the system are upper-triangular there exists, as shown in Example 1, a ﬂag E1 ⊂ E2 ⊂ . . . ⊂ Ep = E of subbundles of rank 1, 2,...,p respectively that are stabilized by the connection ∇. Indeed, a fundamental matrix Y determining the monodromy matrices M1,...,Mn of the system can be represented near each singular point ai in the form Ei Y (z)= Ti(z)(z − ai) , Ei = (1/2πi) ln Mi,

8 where Ti is a meromorphic matrix at the point ai. This matrix can be factored as Ti = Vi Pi, with a holomorphically invertible matrix Vi at ai and an upper-triangular matrix Pi which is a ±1 −1 polynomial in (z − ai) (see, for example, [6, Lemma 1]). Thus the matrix Vi Y is upper- triangular. Let us estimate the degree of each subbundle Ej, j 6 k, noting that in a neighbourhood of each singular point ai the initial system is transformed via a holomorphically invertible gauge transformation in a system with a fundamental matrix of the form

A′ 0 E′ ∗ ′ i i Ui (z) ∗ 0 ∗ 0 ∗ Yi(z)= (z − ai) (z − ai) 0 ∗

′ ′ A′ E′ such that the matrix Yi (z)= Ui (z)(z −ai) i (z −ai) i is a Levelt fundamental matrix for a linear system of j equations with the regular singular point ai. The matrix 1-form of coefficients of j this system in a neighbourhood of ai is a local 1-form of the restriction ∇ of the connection ∇ j ˜1 ˜j to the subbundle E , and the exponents βi ,..., βi of this system (the eigenvalues of the matrix ′ ′ Ai + Ei) form a subset of the exponents of the initial system at ai. Therefore, ˜l Re βi > −1/nk, l = 1,...,j. The degree of the holomorphically trivial vector bundle Ep is equal to zero, and for j 6 k one has: n n n j ′ ′ ′ ′ deg E = resai d ln det Yi (z)= ordai det Ui (z)+ tr(Λi + Ei)= Xi=1 Xi=1 Xi=1 n n j ′ ˜l = ordai det Ui (z)+ Re βi > −j/k > −1. Xi=1 Xi=1 Xl=1 As the degree of a subbundle of a holomorphically trivial vector bundle is non-positive, one has deg Ej = 0, hence all the subbundles E1 ⊂ . . . ⊂ Ek are holomorphically trivial (a subbundle of a holomorphically trivial vector bundle is holomorphically trivial, if its degree is equal to zero, see [11, Lemma 19.16]). Now the assertion of the lemma follows from Lemma 1. N N A matrix A will be called N-resonant, if there are two eigenvalues λ1 6= λ2 such that λ1 = λ2 , that is, 2π |λ | = |λ |, arg λ − arg λ = j, j ∈ {1, 2,...,N − 1}. 1 2 1 2 N Let a group M ⊂ GL(p, C) be generated by matrices M1,...,Mn. If these matrices are sufficiently close to the identity (in the Euclide topology) then the existence of a solvable normal subgroup of finite index in M implies their triangularity, see Theorem 2.7 [13, Ch. 6]. According to the remark following this theorem, the requirement of proximity of the matrices Mi to the identity can be weakened as follows.

Lemma 3. There is a number N = N(p) such that if the matrices M1,...,Mn are not N-resonant, then the existence of a solvable normal subgroup of finite index in M implies their triangularity. Proof of Theorem 1. From the theorem assumptions it follows that the identity component G0 of the differential Galois group G of the system (1) is solvable, hence G0 is a solvable normal subgroup of G of finite index. Then the monodromy group M of this system also has a solvable normal subgroup of finite index, namely M ∩ G0.

9 j As follows from the deﬁnition of the exponents βi of a linear diﬀerential system at a regular j singular point ai, these are connected with the eigenvalues µi of the monodromy matrix Mi by the relation j j µi = exp(2π i βi ). Therefore,

− j j j j 2π Im βi j j µi = exp(2π i(Re βi + i Im βi )) = e (cos(2π Re βi )+ i sin(2π Re βi )),

j and for any N the matrices Mi are non N-resonant by the conditions (14) on the numbers βi . Now the assertion of the theorem follows from Lemma 2 and Lemma 3. As a consequence of Theorem 1 we obtain the following reﬁnement of the Ilyashenko– Khovansky theorem on solvability by quadratures of a Fuchsian system with small residue matrices. j Corollary 1. Let the eigenvalues βi of the residue matrices Bi of the Fuchsian system (2) satisfy the condition 1 Re βj > − , i = 1,...,n, j = 1,...,p, (15) i n(p − 1)

j l and, for each pair βi 6≡ βi (mod Z), i = 1,...,n, one of the conditions (14). Then the solvability of the Fuchsian system (2) by quadratures is equivalent to the simultaneous triangularity of the matrices Bi. Proof. The necessity of simultaneous triangularity is a direct consequence of Theorem 1, since the exponents of the Fuchsian system (2) at ai coincide with the eigenvalues of the residue matrix Bi. Sufficiency follows from a general fact that any linear differential system with an (upper-) triangular coefficient matrix is solvable by quadratures (one should begin with the last equation). Remark 3. The inequalities (15) restricting the real parts of the exponents of the Fuchsian system from below also provide their boundedness from above because of the Fuchs relation n p j i=1 j=1 βi = 0 (see (5)). Namely, P P 1 np − 1 − < Re βj < . n(p − 1) i n(p − 1)

j j In particular, the integer parts ϕi of the numbers Re βi for such a system have to belong to the set {−1, 0, 1}.

Remark 4. If each residue matrix Bi of the Fuchsian system (2) only has one eigenvalue βi, then the solvability of this system by quadratures is also equivalent to the simultaneous triangularity of the matrices Bi (not depending on values of Re βi). Indeed, in this case each 2πiβ monodromy matrix Mi only has one eigenvalue µi = e i , hence is not N-resonant. Then the solvability implies the simultaneous triangularity of the monodromy matrices and existence of a ﬂag E1 ⊂ E2 ⊂ . . . ⊂ Ep = E of subbundles of the holomorphically trivial vector bundle E that are stabilized by the logarithmic connection ∇ (corresponding to the Fuchsian system). n n j Since deg E = i=1 pβi = 0, the degree i=1 jβi of each subbundle E is zero and all these subbundles areP holomorphically trivial. P

10 It is natural to expect that for a general Fuchsian system (with no restrictions on the exponents) solvability by quadratures not necessarily implies simultaneous triangularity of the residue matrices. This is indeed illustrated by the following example of A. Bolibrukh.

Example 2 (A. Bolibrukh [2, Prop. 5.1.1]). There exist points a1, a2, a3, a4 on the Riemann sphere and a Fuchsian system with singularities at these points, whose monodromy matrices are

1 1 1 −1 0 0 −1 1101110  0 −1 −1 000 1   0 −1 1 1 −1 1 −1  0 0 1 122 2 0 0 −1 −1 1 −1 0     M =  0 0 0 110 1  , M =  0001110  , 1   2    0 0 0 011 1   0 0 0 0 −1 −1 1       0 0 0 001 0   0000010       0 0 0 000 −1   000000 −1      1 0 1 0 −1 0 0 1 0 1 −111 0  0 −1 −1 1 −1 1 2   0 −1 1 −200 0  0 0 1 1 −1 1 2 0 0 −1 100 0     M =  0 0 0 −1 1 −1 −2  , M =  0 0 0 −110 1  , 3   4    0 0 0 0 −1 1 1   0 0 0 011 0       0000011   0 0 0 001 1       000000 −1   0 0 0 000 −1      whereas the coeﬃcient matrix of this system is not upper-triangular. Moreover, this system cannot be transformed in an upper-triangular form neither via a constant gauge transformation nor even a meromorphic (rational) one preserving the singular points a1, a2, a3, a4 and the orders of their poles. Thus the residue matrices of this Fuchsian system are not triangular in any basis though the system is solvable by quadratures, since its monodromy group generated by the triangular matrices is solvable. We notice that Corollary 1 does not apply to this example as its exponents cannot satisfy j j j the conditions (15). Indeed, for any exponent βi = ϕi + ρi one has 1 ρj = ln µj, µj ∈ {−1, 1}, i 2πi i i

j hence Re ρi is equal to 0 or 1/2. The inequalities (15) imply (for n = 4, p = 7) 1 Re βj > − , i 24

j hence ϕi is equal to 0 or 1 (see Remark 3). Therefore, the sum of the exponents over all singular points is positive, which contradicts the Fuchs relation.

5 A criterion of solvability for a non-resonant irregular system with small formal exponents

Consider the system (1) with non-resonant irregular singular points a1, . . . , an of Poincar´erank r1, . . . , rn respectively. If the real part of the formal exponents of this system is suﬃciently small, then the following criterion of solvability by quadratures holds.

11 j Theorem 2. Let at each singular point ai the formal exponents λi of the irregular system (1) be pairwise distinct and satisfy the condition 1 Re λj > − , i n(p − 1)

j l and, for every pair (λi , λi), one of the conditions (14). Then this system is solvable by quadratures if and only if there is a constant matrix C ∈ GL(p, C) such that CB(z)C−1 is upper- triangular. Proof. As in Corollary 1, sufficiency does not require a special proof. Let us prove necessity. Consider a fundamental matrix Y of the system (1) and the representation of the differential Galois group G with respect to this matrix. The connection matrices between Y and the 1 1 fundamental matrices Y1 ,...,Yn (the latter were defined at the end of Section 2) denote by P1,...,Pn respectively: 1 Y (z)= Yi (z)Pi, i = 1,...,n. −1 1 Then, as we noted earlier, the monodromy matrices Mi = Pi Mi Pi (i = 1,...,n) with respect to Y belong to the group G. Moreover, as follows from Ramis’ theorem [18]1, the corresponding Stokes matrices

1 −1 1 Ni −1 Ni Ci = Pi Ci Pi, ..., Ci = Pi Ci Pi (i = 1,...,n) also belong to G. Therefore,e by the relatione

e 2πiΛi 1 Ni −1 e = MiCi . . . Ci , Λi = Pi ΛiPi,

e e iee (see (8)), the group G also contains the matrices e2π Λi of formal monodromy. e 2πiΛi 1 Ni n Denote by M the group he , Ci ,..., Ci ii=1 generated by the matrices of formal monodromy and Stokes matrices over all singular points. As follows from the condition of the c e e theorem, the group G possesses the solvable normal subgroup G0 of finite index. Hence the subgroup M ⊂ G possesses the solvable normal subgroup of finite index, M ∩ G0. Since for any N the matrices generating the group M are non N-resonant, according to Lemma 3 they c c are simultaniously reduced to an upper-triangular form by conjugating to some non-degenerated c ie matrix C (non-resonance of the formal monodromy matrices e2π Λi follows from the conditions on the formal exponents, the eigenvalues of the matrices Λ , and is proved as in Theorem 1; for e i the Stokes matrices it follows from their unipotence). We may assume that they are already e upper-triangular (otherwise we would consider the fundamental matrix Y C instead of Y ). As follows from [5, Ch. VIII, §1], the relation e −1 Λi = Pi Λi Pi , e where Λi is an upper-triangular matrix and Λi is a diagonal matrix whose diagonal elements are pairwise distinct, implies that the matrix Pi writes Pi = Di Ri, where Ri is an upper-triangular e −1 matrix (the conjugation Ri Λi Ri annulates all the off-diagonal elements of the matrix Λi) and − D is a permutation matrix for Λ (that is, the conjugation D 1 Λ D permutes the diagonal i e i i i i e elements of the matrix Λi).

1 It is diﬃcult to ﬁnd the original proof of it, but there exist various variants of this theorem and comments to it proposed by the other authors (see [10, Ths. 1, 6], [14], [16, Th. 2.3.11]+[17, Prop. 1.3]).

12 1 Ni In a neighbourhood of each ai we pass from the set of fundamental matrices Yi ,...,Yi , 1 Ni which correspond to the sectors Si ,...,Si , to the fundamental matrices

j j 1 Yi (z)= Yi (z)Pi (in particular, Yi = Y )

e j j+1 e connected to each other in the intersections Si ∩ Si by the relations

j+1 j j Yi (z)= Yi (z)Ci .

e e e j From the decomposition of Pi above and from (6) it follows that the matrices Yi can be written

′ ′ j j Λi Qi(z) j Λi Qi(z) e Yi (z)= Fi (z)(z − ai) e Di Ri = Fi (z)Di(z − ai) e Ri, where e ′ −1 ′ −1 Λi = Di Λi Di, Qi(z)= Di Qi(z) Di are diagonal matrices obtained from the corresponding matrices Λi, Qi(z) by a permutation of j j+1 the diagonal elements. Therefore, in the intersections Si ∩ Si we have the relations

′ ′ ′ ′ j+1 Λi Qi(z) j Λi Qi(z) j Fi (z)Di(z − ai) e Ri = Fi (z)Di(z − ai) e Ri Ci .

1 Ni e Thus in the sectors Si ,...,Si , which form a covering of a punctured neighbourhood of ai, 1 Ni there are holomorphically invertible matrices Fi (z)Di,...,Fi (z) Di respectively such that in j j+1 the intersections Si ∩ Si their quotients − 1 ′ ′ − − ′ − ′ j j+1 Λi Qi(z) j 1 Qi(z) Λi Fi (z)Di Fi (z)Di = (z − ai) e Ri Ci Ri e (z − ai) j e are upper-triangular matrices. As each matrix Fi (z)Di has the same asymptotic expansion j Fi(z)Di in the corresponding Si , there exists a matrix Γi(z) holomorphically invertible at the point a such that all the matrices b i j j Fi (z)=Γi(z)Fi (z)Di, j = 1,...,Ni, are upper-triangular (accordinge to [1, Prop. 3]). In particular,

′ ′ ′ ′ 1 1 Λi Qi(z) 1 Λi Qi(z) Γi(z)Y (z)=Γi(z)Yi (z)=Γi(z)Fi (z)Di(z − ai) e Ri = Fi (z)(z − ai) e Ri is an upper-triangular matrix.e Hence (see Example 1) one has a flageE1 ⊂ E2 ⊂ . . . ⊂ Ep = E of subbundles of rank 1, 2,...,p respectively that are stabilized by the connection ∇. The bounds on the formal exponents imply that these subbundles are holomorphically trivial, whence the assertion of the theorem follows. The proof proceeds as for Lemma 2, there are only two small differences: the first one is that now we deal with formal fundamental matrices of subsystems and their formal exponents, which form a subset of the formal exponents of the initial system at each irregular singular point ai; the second one is the appearence of an exponential factor in a formal fundamental matrix of a subsystem, but the logarithmic differential of such a factor has a zero residue at ai.

13 References

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